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Mirrors > Home > MPE Home > Th. List > cotr3 | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.) |
Ref | Expression |
---|---|
cotr3.a | ⊢ 𝐴 = dom 𝑅 |
cotr3.b | ⊢ 𝐵 = (𝐴 ∩ 𝐶) |
cotr3.c | ⊢ 𝐶 = ran 𝑅 |
Ref | Expression |
---|---|
cotr3 | ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cotr3.a | . . 3 ⊢ 𝐴 = dom 𝑅 | |
2 | 1 | eqimss2i 4044 | . 2 ⊢ dom 𝑅 ⊆ 𝐴 |
3 | cotr3.b | . . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶) | |
4 | cotr3.c | . . . . 5 ⊢ 𝐶 = ran 𝑅 | |
5 | 1, 4 | ineq12i 4211 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅) |
6 | 3, 5 | eqtri 2761 | . . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅) |
7 | 6 | eqimss2i 4044 | . 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 |
8 | 4 | eqimss2i 4044 | . 2 ⊢ ran 𝑅 ⊆ 𝐶 |
9 | 2, 7, 8 | cotr2 14924 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∀wral 3062 ∩ cin 3948 ⊆ wss 3949 class class class wbr 5149 dom cdm 5677 ran crn 5678 ∘ ccom 5681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 |
This theorem is referenced by: (None) |
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